I bred 2 villagers by trading, One was a farmer, the other a leatherworker.
I put the villagers into a farm area for the farmer to harvest the crops and breed with the other villager and he will not harvest the carrots. When I throw either of them carrots they do no pick it up. I have not traded with these villagers so I didn't think their slots would be full. But because they cannot harvest or carry food, they will not breed.